$\sigma$-SCF: A Direct Energy-targeting Method To Mean-field Excited States*

The mean-field solutions of electronic excited states are much less accessible than ground state (e.g.\ Hartree-Fock) solutions. Energy-based optimization methods for excited states, like $\Delta$-SCF, tend to fall into the lowest solution consistent with a given symmetry -- a problem known as ``variational collapse". In this work, we combine the ideas of direct energy-targeting and variance-based optimization in order to describe excited states at the mean-field level. The resulting method, $\sigma$-SCF, has several advantages. First, it allows one to target any desired excited state by specifying a single parameter: a guess of the energy of that state. It can therefore, in principle, find \emph{all} excited states. Second, it avoids variational collapse by using a variance-based, unconstrained local minimization. As a consequence, all states -- ground or excited -- are treated on an equal footing. Third, it provides an alternate approach to locate $\Delta$-SCF solutions that are otherwise hardly accessible by the usual non-aufbau configuration initial guess. We present results for this new method for small atoms (He, Be) and molecules ($\textrm{H}_2$, HF).